Question: does there exist a finite projective plane of order n when n is NOT a prime power and satisfies the Bruck–Ryser–Chowla necessary condition? Known: if n ≡ 1 or 2 (mod 4) and n is not a sum of two squares, no plane exists (Bruck–Ryser–Chowla 1949). If n ≡ 1 mod 4 or n even: open.
The prime-power conjecture (every finite projective plane has prime-power order) was disproved conceptually by computational searches ruling out n = 6 (Tarry 1900) and n = 10 (Lam et al 1989, thousands of CPU hours). No plane of order 12 exists either (2020s computations). But these negative results don't generalise: the Bruck–Ryser conditions allow infinitely many composite orders, and constructing or ruling out planes at those parameters requires fundamentally new ideas.
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